The primary purpose of this research is to develop statistical procedures that are applicable to two-sample data collection situations where either (1) one of the two samples (with a fixed sample size) has previously been collected, but the second sample can be obtained sequentially, or (2) neither sample has been collected, but one of the samples is easy and relatively inexpensive to obtain (for example, "control" observations), while the other sample is costly, risky and/or difficult to collect (for example, "treatment" observations). In such situations we would like to collect a large "control" sample and then to gather the difficult-to-obtain "treatment" observations sequentially, in order to minimize the number of treatment observations which are required in order to reach a statistical decision. The procedures to be developed are nonparametric in nature, thus minimizing the need for assumptions about the form of the distributions underlying the sample observations. Heaviest emphasis will be on the two-sample setting with suspected location differences; however, the use of partially sequential techniques with k-sample or multivariate data and/or differences other than location will be considered.